We derive $C^{1,\sigma}$-estimate for the solutions of a class of non-local elliptic Bellman-Isaacs equations. These equations are fully nonlinear and are associated with infinite horizon stochastic differential game problems involving jump-diffusions. The non-locality is represented by the presence of fractional order diffusion term and we deal with the particular case of $\frac 12$-Laplacian, where the order $\frac 12$ is known as the critical order in this context. More importantly, these equations are not translation invariant and we prove that the viscosity solutions of such equations are $C^{1,\sigma}$, making the equations classically solvable.

L. Silvestre, On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion,, \emph{Advances in Mathematics}, 226 (2011), 2020.
doi: 10.1016/j.aim.2010.09.007.Google Scholar

L. Silvestre, On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion,, \emph{Advances in Mathematics}, 226 (2011), 2020.
doi: 10.1016/j.aim.2010.09.007.Google Scholar